Let \(\varepsilon _1,\dots ,\varepsilon _n\)  denote i.i.d Rademacher random variable independents from \(Y_1,\dots ,Y_n\) .
By the symmetrization lemma (see [1]}),
\(\mathbb {E}\left[\left\Vert \sum _{i=1} ^n Y_i \right\Vert ^{2}\right]&\le 4\mathbb {E}\left[\left\Vert \sum _{i=1} ^n \varepsilon _i Y_i \right\Vert ^{2}\right]= 4\mathbb {E}\left[\sum _{i=1} ^n \left\Vert  Y_i \right\Vert ^{2}\right]=4n\mathbb {E}[\Vert Y\Vert ^2].\) 
